8
18
(K8a
12
)
A knot diagram
1
Linearized knot diagam
7 8 1 2 3 4 5 6
Solving Sequence
1,7 2,4
5 3 6 8
c
1
c
4
c
3
c
6
c
8
c
2
, c
5
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hb + u, −u
3
+ u
2
+ a − 2, u
4
− 2u
3
+ u
2
+ 2u − 1i
I
u
2
= h−u
3
+ 2u
2
+ b − 3u + 1, 4u
3
− 9u
2
+ 3a + 15u − 9, u
4
− 3u
3
+ 6u
2
− 6u + 3i
I
u
3
= hb + a + u + 1, a
2
+ au + 1, u
2
+ u + 1i
I
u
4
= hb + u, 2u
3
+ 4u
2
+ a + 3u − 2, u
4
+ u
3
− 2u + 1i
I
u
5
= h−u
3
− 2u
2
+ b − 2u + 1, 2u
3
+ 3u
2
+ a + 2u − 3, u
4
+ u
3
− 2u + 1i
I
u
6
= hb + u, a − u, u
2
+ u + 1i
I
u
7
= hb, a − 1, u − 1i
I
u
8
= hb − 1, a, u + 1i
I
u
9
= hb − 1, a − 1, u + 1i
I
v
1
= ha, b + 1, v −1i
* 10 irreducible components of dim
C
= 0, with total 26 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I. I
u
1
= hb + u, −u
3
+ u
2
+ a − 2, u
4
− 2u
3
+ u
2
+ 2u − 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
2
=
1
u
2
a
4
=
u
3
− u
2
+ 2
−u
a
5
=
1
−u
3
+ 2u
2
− 1
a
3
=
u
3
− u
2
− u + 2
−u
a
6
=
−u
3
+ 2u
2
− 2
u
3
− u
2
+ 1
a
8
=
−u
−u
2
+ 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
3
− 4u
2
+ 4u + 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
7
u
4
− 2u
3
+ u
2
+ 2u − 1
c
2
, c
4
, c
6
c
8
u
4
+ 2u
3
+ u
2
− 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
y
4
− 2y
3
+ 7y
2
− 6y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.883204
a = 0.531010
b = 0.883204
−1.71901 −5.40880
u = 0.468990
a = 1.88320
b = −0.468990
1.71901 5.40880
u = 1.20711 + 0.97832I
a = −0.207107 + 0.978318I
b = −1.20711 − 0.97832I
− 12.3509I 0. + 7.82655I
u = 1.20711 − 0.97832I
a = −0.207107 − 0.978318I
b = −1.20711 + 0.97832I
12.3509I 0. − 7.82655I
5
II.
I
u
2
= h−u
3
+2u
2
+b−3u +1, 4u
3
−9u
2
+3a+15u −9, u
4
−3u
3
+6u
2
−6u+3i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
2
=
1
u
2
a
4
=
−
4
3
u
3
+ 3u
2
− 5u + 3
u
3
− 2u
2
+ 3u − 1
a
5
=
−
1
3
u
3
− 1
−1
a
3
=
−
1
3
u
3
+ u
2
− 2u + 2
u
3
− 2u
2
+ 3u − 1
a
6
=
4
3
u
3
− 2u
2
+ 4u − 1
−u
2
+ 2u − 2
a
8
=
−
2
3
u
3
+ u
2
− 2u + 1
−u
3
+ 2u
2
− 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 8u
3
− 16u
2
+ 24u − 6
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
4
− 3u
3
+ 6u
2
− 6u + 3
c
2
, c
4
, c
6
c
8
u
4
− u
3
+ 2u + 1
c
3
, c
7
(u
2
+ u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
4
+ 3y
3
+ 6y
2
+ 9
c
2
, c
4
, c
6
c
8
y
4
− y
3
+ 6y
2
− 4y + 1
c
3
, c
7
(y
2
+ y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.851597 + 0.632502I
a = 0.25679 − 1.42811I
b = 0.500000 + 0.866025I
1.64493 − 4.05977I 6.00000 + 6.92820I
u = 0.851597 − 0.632502I
a = 0.25679 + 1.42811I
b = 0.500000 − 0.866025I
1.64493 + 4.05977I 6.00000 − 6.92820I
u = 0.64840 + 1.49853I
a = −0.256789 + 0.303939I
b = 0.500000 − 0.866025I
1.64493 + 4.05977I 6.00000 − 6.92820I
u = 0.64840 − 1.49853I
a = −0.256789 − 0.303939I
b = 0.500000 + 0.866025I
1.64493 − 4.05977I 6.00000 + 6.92820I
9
III. I
u
3
= hb + a + u + 1, a
2
+ au + 1, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
2
=
1
−u − 1
a
4
=
a
−a − u − 1
a
5
=
au + a − u − 1
−1
a
3
=
−u − 1
−a − u − 1
a
6
=
−au − a + u
au
a
8
=
−2au − a
−a + u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −8u + 2
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u
2
+ u + 1)
2
c
2
, c
4
, c
6
c
8
u
4
− u
3
+ 2u + 1
c
3
, c
7
u
4
− 3u
3
+ 6u
2
− 6u + 3
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y
2
+ y + 1)
2
c
2
, c
4
, c
6
c
8
y
4
− y
3
+ 6y
2
− 4y + 1
c
3
, c
7
y
4
+ 3y
3
+ 6y
2
+ 9
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0.148403 + 0.632502I
b = −0.64840 − 1.49853I
1.64493 + 4.05977I 6.00000 − 6.92820I
u = −0.500000 + 0.866025I
a = 0.35160 − 1.49853I
b = −0.851597 + 0.632502I
1.64493 + 4.05977I 6.00000 − 6.92820I
u = −0.500000 − 0.866025I
a = 0.148403 − 0.632502I
b = −0.64840 + 1.49853I
1.64493 − 4.05977I 6.00000 + 6.92820I
u = −0.500000 − 0.866025I
a = 0.35160 + 1.49853I
b = −0.851597 − 0.632502I
1.64493 − 4.05977I 6.00000 + 6.92820I
13
IV. I
u
4
= hb + u, 2u
3
+ 4u
2
+ a + 3u − 2, u
4
+ u
3
− 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
2
=
1
u
2
a
4
=
−2u
3
− 4u
2
− 3u + 2
−u
a
5
=
−u
3
− 2u
2
− 2u
−1
a
3
=
−2u
3
− 4u
2
− 4u + 2
−u
a
6
=
−5u
3
− 8u
2
− 4u + 8
−u
3
− 2u
2
− u + 2
a
8
=
−4u
3
− 7u
2
− 5u + 5
−u
3
− 2u
2
− u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 8u
3
+ 16u
2
+ 8u − 18
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
7
u
4
+ u
3
− 2u + 1
c
2
, c
6
u
4
+ 3u
3
+ 6u
2
+ 6u + 3
c
4
, c
8
(u
2
− u + 1)
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
7
y
4
− y
3
+ 6y
2
− 4y + 1
c
2
, c
6
y
4
+ 3y
3
+ 6y
2
+ 9
c
4
, c
8
(y
2
+ y + 1)
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.621964 + 0.187730I
a = −1.62196 − 1.91978I
b = −0.621964 − 0.187730I
−1.64493 − 4.05977I −6.00000 + 6.92820I
u = 0.621964 − 0.187730I
a = −1.62196 + 1.91978I
b = −0.621964 + 0.187730I
−1.64493 + 4.05977I −6.00000 − 6.92820I
u = −1.12196 + 1.05376I
a = 0.121964 + 0.678295I
b = 1.12196 − 1.05376I
−1.64493 + 4.05977I −6.00000 − 6.92820I
u = −1.12196 − 1.05376I
a = 0.121964 − 0.678295I
b = 1.12196 + 1.05376I
−1.64493 − 4.05977I −6.00000 + 6.92820I
17
V. I
u
5
= h−u
3
− 2u
2
+ b − 2u + 1, 2u
3
+ 3u
2
+ a + 2u − 3, u
4
+ u
3
− 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
2
=
1
u
2
a
4
=
−2u
3
− 3u
2
− 2u + 3
u
3
+ 2u
2
+ 2u − 1
a
5
=
1
2u
3
+ 4u
2
+ 2u − 2
a
3
=
−u
3
− u
2
+ 2
u
3
+ 2u
2
+ 2u − 1
a
6
=
u
3
+ 2u
2
+ 2u − 1
u
3
+ u
2
+ u − 2
a
8
=
−u
−2u
3
− 2u
2
− u + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 8u
3
+ 16u
2
+ 8u − 18
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
7
u
4
+ u
3
− 2u + 1
c
2
, c
6
(u
2
− u + 1)
2
c
4
, c
8
u
4
+ 3u
3
+ 6u
2
+ 6u + 3
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
7
y
4
− y
3
+ 6y
2
− 4y + 1
c
2
, c
6
(y
2
+ y + 1)
2
c
4
, c
8
y
4
+ 3y
3
+ 6y
2
+ 9
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 0.621964 + 0.187730I
a = 0.35160 − 1.49853I
b = 1.12196 + 1.05376I
−1.64493 − 4.05977I −6.00000 + 6.92820I
u = 0.621964 − 0.187730I
a = 0.35160 + 1.49853I
b = 1.12196 − 1.05376I
−1.64493 + 4.05977I −6.00000 − 6.92820I
u = −1.12196 + 1.05376I
a = 0.148403 − 0.632502I
b = −0.621964 + 0.187730I
−1.64493 + 4.05977I −6.00000 − 6.92820I
u = −1.12196 − 1.05376I
a = 0.148403 + 0.632502I
b = −0.621964 − 0.187730I
−1.64493 − 4.05977I −6.00000 + 6.92820I
21
VI. I
u
6
= hb + u, a − u, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
2
=
1
−u − 1
a
4
=
u
−u
a
5
=
−1
0
a
3
=
0
−u
a
6
=
−1
u + 1
a
8
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u − 4
22
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
7
u
2
+ u + 1
c
2
, c
4
, c
6
c
8
u
2
− u + 1
23
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
y
2
+ y + 1
24
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −0.500000 + 0.866025I
b = 0.500000 − 0.866025I
4.05977I 0. − 6.92820I
u = −0.500000 − 0.866025I
a = −0.500000 − 0.866025I
b = 0.500000 + 0.866025I
− 4.05977I 0. + 6.92820I
25
VII. I
u
7
= hb, a − 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
1
a
2
=
1
1
a
4
=
1
0
a
5
=
0
−1
a
3
=
1
0
a
6
=
−1
1
a
8
=
0
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
26
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
8
u − 1
c
3
, c
7
u
27
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
8
y − 1
c
3
, c
7
y
28
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 0
1.64493 6.00000
29
VIII. I
u
8
= hb − 1, a, u + 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
−1
a
2
=
1
1
a
4
=
0
1
a
5
=
1
2
a
3
=
1
1
a
6
=
0
−1
a
8
=
1
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
30
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
7
, c
8
u + 1
c
2
, c
6
u
31
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
7
, c
8
y − 1
c
2
, c
6
y
32
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
8
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = 1.00000
−1.64493 −6.00000
33
IX. I
u
9
= hb − 1, a − 1, u + 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
−1
a
2
=
1
1
a
4
=
1
1
a
5
=
1
1
a
3
=
2
1
a
6
=
1
0
a
8
=
1
0
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
34
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
u + 1
c
4
, c
8
u
35
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
y − 1
c
4
, c
8
y
36
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
9
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 1.00000
b = 1.00000
−1.64493 −6.00000
37
X. I
v
1
= ha, b + 1, v − 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
1
0
a
2
=
1
0
a
4
=
0
−1
a
5
=
−1
−1
a
3
=
−1
−1
a
6
=
1
1
a
8
=
2
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
38
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
c
2
, c
3
, c
4
c
6
, c
7
, c
8
u − 1
39
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
c
2
, c
3
, c
4
c
6
, c
7
, c
8
y − 1
40
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −1.00000
1.64493 6.00000
41
XI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
7
u(u − 1)(u + 1)
2
(u
2
+ u + 1)
3
(u
4
− 3u
3
+ 6u
2
− 6u + 3)
· (u
4
− 2u
3
+ u
2
+ 2u − 1)(u
4
+ u
3
− 2u + 1)
2
c
2
, c
4
, c
6
c
8
u(u − 1)
2
(u + 1)(u
2
− u + 1)
3
(u
4
− u
3
+ 2u + 1)
2
· (u
4
+ 2u
3
+ u
2
− 2u − 1)(u
4
+ 3u
3
+ 6u
2
+ 6u + 3)
42
XII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
y(y − 1)
3
(y
2
+ y + 1)
3
(y
4
− 2y
3
+ 7y
2
− 6y + 1)
· (y
4
− y
3
+ 6y
2
− 4y + 1)
2
(y
4
+ 3y
3
+ 6y
2
+ 9)
43
